A117621 Number of double-perfect partitions of [1..n].

0, 1, 1, 1, 1, 1, 3, 1, 3, 2, 3, 1, 7, 1, 3, 3, 6, 1, 8, 1, 7, 3, 3, 1, 17, 2, 3, 4, 7, 1, 13, 1, 12, 3, 3, 3, 24, 1, 3, 3, 17, 1, 13, 1, 7, 8, 3, 1, 40, 2, 8, 3, 7, 1, 20, 3, 17, 3, 3, 1, 41, 1, 3, 8, 24, 3, 13, 1, 7, 3, 13, 1, 68, 1, 3, 8, 7, 3, 13, 1, 40, 8, 3, 1, 41, 3, 3, 3, 17, 1, 44, 3, 7, 3, 3, 3

Offset: 1

Author

N. J. A. Sloane, Apr 07 2006

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..20000
• HoKyu Lee, Double perfect partitions, Discrete Math., 306 (2006), 519-525.

Crossrefs

Cf. A002033.

Programs

Maple

f:=proc(n) option remember; local t1,m,nm1,mm1; nm1:=n-1; if n <= 1 then RETURN(0); elif n <= 5 then RETURN(1); else t1:=0; for m from 2 to n-1 do mm1:=m-1; if nm1 mod mm1 = 0 then t1:=t1+f(m); fi; od; RETURN(t1); fi; end;
# second Maple program:
with(numtheory):
a:= proc(n) option remember; `if`(n<6, [0, 1$4][n],
      add(a(k+1), k=divisors(n-1) minus {n-1}))
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Jul 09 2015

Mathematica

a[n_] := a[n] = If[n<6, {0, 1, 1, 1, 1}[[n]], Sum[a[k+1], {k, Divisors[n-1] ~Complement~ {n-1}}]]; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Jul 15 2015, after Alois P. Heinz *)

Formula

a(1)=0; a(n)=1 for n=2..5; a(n) = Sum_{m=2..n-1, m-1|n-1} a(m) for n >= 6.